Bitter rivals Liverpool clinched their first title in 30 years this campaign but United's No. 1 expects his team to mount more of a challenge next season.
"We are Manchester but it's been too long without competing for important things and I think from now on, starting next season, we have to compete for those big titles," De Gea told DAZN Espana.
De Gea won his first and only Premier league title in the 2012-13 campaign, which was Sir Alex Ferguson's last season in charge of the club.
United have not enjoyed much success since under David Moyes, Louis van Gaal and Jose Mourinho.
"In the end, he [Ferguson] did so much, it was all so good, so many league titles, so much success, that obviously the transition had to be difficult," De Gea added.
"I know what it is like to win a Premier League with this club. I know what it means to win something big. I want people to know again what this club is and I want to help with that. And I hope I can stay many years here."
With three games remaining, United are level on 59 points with fourth-placed Leicester City. The last Champions League place could be decided on the final day of the campaign when both teams face each other at the King Power stadium.
"I'm very comfortable at this club, it's already been 400 games, which proves that this club is incredible and that we can continue to be at the top again, win things and play another 400 more," De Gea said.
"We are very well and very happy with the coach. It seems that the team is in a good dynamic, on the right path and we have to give our all and continue like this."